Stern-Brocot trees in the periodicity of mixed-mode oscillations

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Freire and Gallas
Stern–Brocot trees in the periodicity
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PAPER
Stern–Brocot trees in the periodicity of mixed-mode oscillations
Joana G. Freire ab and Jason A. C. Gallas* abc
Received 3rd December 2010, Accepted 13th January 2011
DOI: 10.1039/c0cp02776f
We investigate the distribution of mixed-mode oscillations in the control parameter space
for two paradigmatic chemical models: a three-variable fourteen-parameter model of the
Belousov–Zhabotinsky reaction and a three-variable four-parameter autocatalator. For both
systems, several high-resolution phase diagrams show that the number of spikes of their
mixed-mode oscillations emerges consistently organized in a surprising and unexpected
symmetrical way, forming Stern–Brocot trees. The Stern–Brocot tree is more general and
contains the Farey tree as a subtree. We conjecture the Stern–Brocot hierarchical organization
to be the archetypal skeleton underlying several systems displaying mixed-mode oscillations.
1. Introduction
Mixed-mode oscillations (MMOs) are complex oscillatory
patterns consisting of trains of small amplitude oscillations
followed by large excursions of relaxation type. During the last
30 years or so, MMOs were observed profusely in experiments
and models of prototypic chemical systems. 1–24 In the
literature, MMOs were also called alternating periodic-chaotic
sequences. 14,15 For a recent survey about the properties,
prospection and use of MMOs in several fields see ref. 25.
Although MMOs were already investigated abundantly,
they were analyzed most frequently by considering the
dynamics observed along a single or a few scattered oneparameter
sections of parameter spaces that are normally of
very high dimensions. Sometimes, a few bifurcation curves
were obtained using numerical continuation techniques. It
seems then natural to ask if restricted sections of the
high-dimensional parameter spaces are sufficient for an
unambiguous characterization of MMO cascades. Here we
show that the periodicity of MMO cascades still harbors
unsuspected and far-reaching organizational features.
The aim of this paper is to present a detailed investigation of
the unfolding of MMO cascades as observed when two
parameters are tuned finely in the control space of two
representative examples of chemical dynamics, namely the
Belousov–Zhabotinsky reaction 4,5 and in one of the
commonly exploited model schemes based on isothermal
autocatalysis. 6,7 This is done by numerically computing
high-resolution planar phase diagrams for these systems.
a Instituto de Fı´sica, Universidade Federal do Rio Grande do Sul,
91501-970 Porto Alegre, Brazil. E-mail: jgallas@if.ufrgs.br
b Departamento de Matema´tica, Centro de Estruturas Lineares e
Combinato´rias, Universidade de Lisboa, Lisboa, Portugal
c Departamento de Fı´sica, Universidade Federal da Paraı´ba,
58051-970 Joa˜o Pessoa, Brazil
The main result reported here is the discovery of a
surprisingly new hierarchical organization for MMOs in
parameter space of both examples mentioned. In contrast to
what is presently known, in both models we find MMOs to
emerge organized in perfect agreement with the so-called
Stern–Brocot tree, 26,27 not according to the familiar Farey
tree. 28 The Stern–Brocot trees are more general than Farey
trees and include them as subtrees. 29,30 Stern–Brocot trees may
be recognized contemplating the unfolding of oscillations in
several two-parameter sections of the control parameter space.
Before proceeding, we remark that we also found Stern–
Brocot trees in other popular models of MMOs. Here,
however, we wish to focus on the aforementioned pair of
representative models, namely a realistic Belousov–Zhabotinsky
proxy and the autocatalator.
Our motivation for this work arises from the fact that the
period-adding sequences commonly found in MMOs systems
display a striking resemblance with aspects of sequences
observed recently in rather distinct scenarios connected with
certain ‘‘periodicity hubs’’. 31–34 When projected in planar
phase diagrams, multidimensional hub manifolds produce
remarkable networks of points responsible for organizing very
regularly the dynamics around quite wide portions of the
parameter space. 34 For us, systems displaying MMOs are
particularly attractive to investigate and probe details
associated with the nature of certain intricate reinjection
mechanisms, homoclinic or not, causing periodicity hubs 33,34
and of reinjections arising in multiple-timescale systems.
Such mechanisms need to be investigated because they play
important roles in the generation of a radically distinct class of
stability spirals than those presently known, a class for which
not even a local adequate mathematical framework is available
at present. 35
In the next section we review briefly a realistic model of the
Belousov–Zhabotinsky reaction and introduce what we call
‘‘isospike diagrams’’ illustrating how the number of spikes of
This journal is c the Owner Societies 2011 Phys. Chem. Chem. Phys., 2011, 13, 12191–12198 12191

periodic oscillations auto-organizes in five distinct sections of
their control parameter space. Section 3 describes the
Stern–Brocot organization arguing that this is the hierarchical
organization of the MMOs in the Belousov–Zhabotinsky
reaction. Section 4 presents results for the autocatalator model
providing independent corroboration of the Stern–Brocot
organization. Although isospike diagrams of both models
display the same hierarchical Stern–Brocot structure, they
are easier to be recognized in the autocatalator, in the
sense that autocatalator requires fewer phase diagrams
to display the organization. Finally, Section 5 summarizes
our conclusions.
2. Isospike diagrams for a realistic BZ model
The Belousov–Zhabotinsky reaction is a classical dynamical
system for which several mathematical models have been
developed over the years with an ever increasing degree of
reality. 16–20 Here, we consider a model suggested by Gyo¨ rgyi
and Field 4 that contains 14 parameters that may be tuned to
produce rich dynamical scenarios. 5 It is governed by three
differential equations:
dx
dt ¼ T 0 k 1 HY 0 x~y þ k 2 AH 2 Y 0
~y 2k 3 X 0 x 2
X 0
þ 0:5k 4 A 0:5 H 1:5 X 0:5
0 ðC Z 0 zÞx 0:5
0:5k 5 Z 0 xz k f xg;
dz
dt ¼ T 0 k 4 A 0:5 H 1:5 X0
0:5 C
Z 0
ak 6 V 0 zv bk 7 Mz k f zg;
z
x 0:5
dv
dt ¼ T Y 0
0 2k 1 HX 0 x~y þ k 2 AH 2 Y 0
~y
V 0 V 0
X 2
0
þ k 3 x 2 ak 6 Z 0 zv k f v ;
V 0
where we abbreviated 5
k 5 X 0 xz
x X X 0
; y Y Y 0
; z Z Z 0
; v V V 0
; t t
T 0
;
X [HBrO 2 ], A [BrO 3 ],
Y [Br ],
Z [Ce(IV)], H [H + ],
C [Ce(III)] + [Ce(IV)],
V [BrMA], M [CH 2 (COOH) 2 ],
and where X 0 = k 2 AH 2 /k 5 , Y 0 =4X 0 ,
Z 0 ¼ CA
40M ; V 0 ¼ 4 AHC
M 2 ; T 1
0 ¼
10k 2 AHC ;
~y ¼
ak 6 Z 0 V 0 zv
ðk 1 HX 0 x þ k 2 AH 2 þ k f ÞY 0
:
ð1Þ
ð2Þ
ð3Þ
ð4Þ
ð5Þ
ð6Þ
Table 1 Numerical values of rate constants and parameters fixed in
our simulations, taken from ref. 4, in the same units
k 1 =4 10 6 k 2 = 2.0
k 3 =3 10 3 k 4 = 55.2
k 5 =7 10 3 K 6 = 0.09
k 7 = 0.23 k f = 3.9 10 4
A = 0.1 C = 8.33 10 4
H = 0.26 M = 0.25
a = 666.7 b = 0.3478
The quantities a and b are historical artifacts and have no
chemical significance. 5 All parameters appearing in the
equations above are collected in Table 1, along with their
basic numerical values. Further details about this model are
available in the literature. 4,5
Fig. 1 presents a pair of phase diagrams that we computed
for eqn (1)–(3). In both diagrams, the parameter region shown
was divided into a mesh of 1200 1200 equally spaced values.
For each parameter pair, we integrated eqn (1)–(3) using a
standard fourth-order Runge–Kutta algorithm with fixed
time-step h = 2 10 6 . The first 2.5 10 6 steps were
discarded as transient. During the next 2.5 10 6 steps we
searched for the local maxima of the variable displaying the
largest number of spikes within a period, x for the BZ reaction.
The number of spikes was then plotted in what we call isospike
diagrams, with the help of colors, as indicated in the diagrams.
As it is known, the number of spikes of each independent
variable is not uniformly equal. 31
Computations were always started from the lowest value
of k 2 from the arbitrary initial condition (x 0 , z 0 , v 0 ) =
(0.046, 0.898, 0.846) and continued by following the attractor,
namely, by using the values of x, z, v obtained at the end of one
integration at a given k 2 to start a new calculation after
incrementing k 2 infinitesimally. This is a standard way of
generating bifurcation and Lyapunov diagrams, and the
rationale behind it is that, generically, basins of attraction
do not change significantly upon slight changes of parameters,
thereby ensuring a relatively smooth unfolding of the quantity
being investigated as parameters evolve. Lyapunov exponents
were computed in a similar way, as described in ref. 5.
Fig. 1a shows an isospike diagram, discriminating with
colors parameter domains characterized by periodic
oscillations having an identical number of spikes within a
period. Such diagram is a sort of generalized isoperiodic
diagram, 36–38 useful for visualizing the organization of periodic
phases. Fig. 1a displays the 14 lowest periods using the 14
colors indicated by the colorbar, recycling them ‘‘mod 14’’ for
higher periods. Black is used to represent parameters leading
to non-periodic oscillations, i.e. to chaos. This same scale is
used in similar figures below where, additionally, white is used
to mark fixed-points (i.e. non-oscillatory solutions).
In Fig. 1a it is possible to recognize a dense succession of
curved stripes indicating that periodic oscillations with an ever
increasing number of spikes exist symmetrically on both sides
of the green stripe (which marks parameters leading to
periodic oscillations with two spikes in a period). On both sides
of the green 2-spike phase one may recognize the unfolding of
the doubling cascade ending in chaos (black). Next comes a
large region characterized by three spikes, together with its
doubling cascade, ending once again in chaos (black).
12192 Phys. Chem. Chem. Phys., 2011, 13, 12191–12198 This journal is c the Owner Societies 2011

Fig. 1 Phase diagrams for the Belousov–Zhabotinsky reaction. (a) Isospike diagram emphasizing periodic phases. Chaotic phases are shown in
black. The white box indicates the location of the lower ‘‘armpit’’ of the 2-spike phase. The three vertical lines indicate the parameters investigated
in the bifurcation diagrams shown in Fig. 2. Colors and numbers mark the number of peaks in a period of x, as indicated by the colorbar. Colors
are used ‘‘mod 14’’, i.e. we recycle the same colors for higher periods. Period-adding cascades are easily recognizable along a continuum of
one-parameter, codimension one, lines. (b) Lyapunov phase diagram emphasizing chaotic phases. Although generated in distinct ways, the
dynamical phases predicted by both diagrams agree with each other. Each panel displays results from the phase-space analysis of the dynamics for
1200 2 = 1.44 10 6 individual parameter points.
Fig. 1b displays a Lyapunov phase diagram 34,39–41 corresponding
to the panel shown in Fig. 1a. The emphasis in the
Lyapunov phase diagram is on enhancing the intricate and
large chaotic phase, i.e. making chaos more easily visible.
Colors denote parameters leading to chaos, i.e. positive
exponents, while darker shadings mark periodicity, as
indicated by the color bar. The color scale for the Lyapunov
exponents is linear on both sides of zero but not uniform from
negative to positive extrema. Note that although Fig. 1a and b
are obtained using two very distinct algorithms, they both
agree: the boundaries between chaotic and periodic regions
delimited by Lyapunov exponents coincide with the
boundaries obtained by counting spikes.
Fig. 1a manifests a clear structural organization of the
isospike phases and it is natural to ask about how this ordering
further develops for higher periods. The large blue phase in
Fig. 1a corresponds to 1-spike oscillations. When moving to
the left from this 1-spike phase one meets the green 2-spike
phase corresponding to a period doubling of the 1-spike
oscillations. Moving further to the left one sees that the
2-spike phase develops two distinct ‘‘armpits’’, i.e. two phases
characterized by period doubling cascades that accumulate in
a chaotic phase, represented in black in Fig. 1a [in yellow in
Fig. 1b]. The white box in Fig. 1a indicates the location of the
lower armpit of the 2-spike phase. Although each chaotic
phase contains a myriad of smaller isospike phases, the largest
periodic phases embedded in the chaotic phases of the 2-spike
armpits are a symmetric pair of 3-spike phases, as indicated by
the labels ‘‘3’’ in the figure. Each one of these 3-spike phases
develops its own symmetric pair of armpits. To see how this
multiplication process develops is difficult in the scale of
Fig. 1a. However, the structure may be recognized resorting
to several magnifications and bifurcation diagrams of specific
portions of the phase diagram (not shown here).
In this way we recognize that the general picture underlying
the hierarchical process at hand is the emergence of an infinite
cascade of armpit pairs appearing in a definite order which,
albeit strongly distorted in the figures, display a mirror
symmetry with respect to the central 2-spike green domain.
This central domain seems to organize the whole structure as
periods grow without bound.
Fig. 2 presents bifurcation diagrams showing the evolution
of the variable x (the normalized concentration X = [HBrO 2 ])
along the three vertical lines shown in Fig. 1a. The bifurcation
diagrams contain numbers to help identify the location of
some of the major windows of periodic behavior and to
facilitate comparison with the planar windows depicted in
Fig. 1a. The bifurcation diagrams emphasize the fact that it
is very hard to grasp the hierarchical ordering of the MMOs
based solely on a few isolated diagrams.
Fig. 3 shows four additional parameter sections of the
14-dimensional parameter space of the Belousov–Zhabotinsky
reaction. As it is clear from this figure, all sections display the
same hierarchical organization found for Fig. 1a. It is
important to realize that the parameters used in all figures
here do not reflect abstract choices but, instead, are
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Fig. 2 Bifurcation diagrams showing the evolution of x (i.e. of the
normalized concentration X = [HBrO 2 ]) along the three vertical lines
shown in Fig. 1a. Numbers indicate the number of spikes characterizing
the windows containing them. The Belousov–Zhabotinsky model has
an accumulation of spikes close to zero as k 4 increases. Bifurcations
along (a) k 2 = 1.1; (b) k 2 = 1.5; (c) k 2 = 2.0.
parameters derived from real experiments. 4 This is why our
cuts produce rather asymmetric trees. We believe to be
possible to optimize parameters so as to generate much more
symmetric parameter sections. Such search, however, is
expected to be computationally very time-consuming and will
not be pursued at this point.
In Fig. 4 we summarize schematically the first few levels of
the hierarchical organization of MMOs observed in the five
sections of the parameter space of the Belousov–Zhabotinsky
reaction, shown in Fig. 1 and 3. To emphasize the symmetry of
the hierarchical structure, in Fig. 4 we compensated shifts and
shear normally present in phase diagrams. The fine structure
of the hierarchical development of MMOs is of course rather
complex but Fig. 4 certainly captures its Leitmotiv.
3. The tree of Stern and Brocot
The correlation between the MMOs observed progressively in
parameter space is traditionally investigated by establishing a
one-to-one correspondence between these oscillations and an
ordered set of rational numbers. To do this it is usual to start
by first establishing a taxonomy for MMOs by introducing
symbols like L s , where L and s refer, respectively, to the
number of ‘‘large’’ and ‘‘small’’ amplitude excursions recorded
in the time evolution of one of its variables.
A well-known ordering of rationals is generated by
assigning to a given pair p/q and p 0 /q 0 of rationals an intermediary
‘‘mediant’’ rational (p + p 0 )/(q + q 0 ). Since the
number of spikes in a period is defined by an integer, not by
a rational number, we consider ‘‘derived trees’’ formed by
simply summing p and q of familiar trees used in number
theory to represent sequences of rationals. Fig. 5 compares the
sequences of rationals generated by Stern–Brocot 29 and by
Farey 28 sequences with the sum trees derived from them by
simply adding numerator and denominator of the rationals.
As it is clear from Fig. 5, the spike ordering of the MMOs in
Fig. 1a does not correspond to the one generated by the Farey
tree but it is in perfect agreement with the integers in the
Stern–Brocot sum tree. This ‘‘good’’ tree was devised independently
in 1858 by Moritz Stern 26 and in 1861 by Achille
Brocot. 27 Stern was a German mathematician and Brocot a
French clockmaker. The latter used this tree to design systems
of gears with a gear ratio close to some desired value by
finding a ratio of numbers near that value. The Stern–Brocot
and Farey trees are generated by the same arithmetic principle.
However, Stern–Brocot trees are more general than Farey
trees and include them as subtrees. 29,30
The Stern–Brocot sequence differs from the Farey sequence
in two basic ways: 29 it eventually includes all positive
rationals, not just the rationals within the interval [0,1], and
at the nth step all mediants are included, not only the ones with
denominator equal to n. The Farey sequence of order n may be
found by an in-order traversal of the left subtree of the
Stern–Brocot tree, backtracking whenever a number with
denominator greater than n is reached. ‘‘But we had better
not discuss the Farey series any further, because the entire
Stern-Brocot tree turns out to be even more interesting.’’ 29
Two factors are important to identify the Stern–Brocot tree:
first, one needs to sweep finely two parameters simultaneously
and, second, the tree is made visible by the isospike diagrams
introduced here. We find the total number of spikes to
be a more reliable indicator than the standard large–small
L s labeling and the associated ‘‘winding numbers’’
W = s/(L + s) derived from such labeling. These quantities
turn out to be rather ambiguous when tuning two parameters
on a finely spaced mesh. For instance, the attribution of the
labels 1 0 and 0 1 is ambiguous from the outset as also is the set
of multiple labels which are possible in sequences of spikes
with comparable amplitudes evolving as parameters are changed
slightly, due to difficulties in distinguishing ‘‘large’’ from
‘‘small’’. Furthermore, the Farey tree has been frequently
identified in the literature on the basis of ‘‘devil staircases’’
constructed with the aforementioned winding numbers, despite
ambiguities and despite the fact that L s and the infinite sequence
(nL) (ns) for n = 2,3,... share identical winding numbers.
Farey trees have been identified in earlier work on the basis
of the analysis of a single chemical variable. It is important to
realize that although phase diagrams obtained by using just
one of the dynamical variables are easier to obtain and
12194 Phys. Chem. Chem. Phys., 2011, 13, 12191–12198 This journal is c the Owner Societies 2011

Fig. 3 Examples of Stern–Brocot trees observed in four additional sections of the 14-dimensional parameter space of the Belousov–Zhabotinsky
reaction. The relative invariance of the diagrams seems to imply the existence of a highly symmetric hyper-manifold in parameter space. White
regions represent fixed-point (non-oscillating) solutions. The white–blue boundaries are lines of Hopf bifurcations. The ‘‘fountains of chaos’’
described in ref. 5 are here identified as manifestations of the Stern–Brocot order defined in Fig. 4 and 5 below. Each panel displays results from the
phase-space analysis of the dynamics for 1200 2 = 1.44 10 6 individual parameter points.
Fig. 4 Schematic representation of the first stages of the hierarchical
unfolding of isospike cascades observed in MMOs. The black regions
between isospike phases represent chaotic phases of non-homogeneous
chaos, 34 reached after cascades of period doubling bifurcations. The
‘‘armpits’’ of the main 2-spike phase contain a pair of 3-spike phases,
each one containing their own pair of armpits with isospike phases,
with this hierarchical organization repeating ad infinitum, showing a
mirror symmetry with respect to the central 2-spike domain. As shown
in Fig. 5 below, the sequence of periods here corresponds to that
generated by the Stern–Brocot tree.
generally reliable, the final diagram might depend on the choice
of the variable. For instance, identical isospike diagrams are
obtained when counting spikes from either x(t)orz(t) in eqn (1)
and (2). The oscillations of v(t) in eqn (3) are less representative
than the ones from the former two variables. This is so because
the number of spikes of individual variables governing
dynamical systems evolves independently from each other. 31
When constructing isospike diagrams, we always use one of the
variables that displays the greatest temporal variation, i.e. the
greatest number of spikes within a period. In any case,
dependencies may be eliminated by using vector quantities
defined by taking into consideration all components governing
the flow, a superfluous step for our present purposes. Be it as it
may, eventual dependencies need to be ascertained on a caseby-case
basis by studying all dynamical variables involved.
We emphasize that isospike diagrams may be used equally
well to discriminate between a Stern–Brocot and a Farey tree.
For any given system, its specific tree may be read directly
from the organization and the sequencing seen in the
computed isospike phases. Note, however, that isospike
diagrams provide no information about the relative
magnitudes or about the sequence of oscillations within a
period. Isospike diagrams work as useful complements to the
traditional tools, like, e.g., time-evolution plots and bifurcation
diagrams, being certainly not a substitute for any of them.
This journal is c the Owner Societies 2011 Phys. Chem. Chem. Phys., 2011, 13, 12191–12198 12195

Fig. 5 Top row: tree of periods obtained by summing numerator and denominator of the fractions in the Stern–Brocot and Farey trees. Bottom
row: schematic representation of the Stern–Brocot and the Farey trees. The sequence of periods derived from the Stern–Brocot tree reproduces the
hierarchy of the periodic phases in Fig. 1a and 3. The arithmetic process responsible for the growth of both trees is the same.
4. Stern–Brocot tree in the autocatalator model
To check the generality of the Stern–Brocot spike-ordering we
performed an additional experiment, investigating the ordering
of MMOs for a paradigmatic autocatalator defined by the
equations 6,7
dA
dt ¼ mðC þ kÞ AðB2 þ 1Þ; ð7Þ
s dB
dt ¼ AðB2 þ 1Þ B; ð8Þ
d dC ¼ B C; ð9Þ
dt
where A, B, C are dimensionless concentrations, and m, k, s, d
are freely tunable control parameters.
Fig. 6 presents phase diagrams demonstrating clearly
that the hierarchical structure of MMOs for the autocatalator
coincides with the one found earlier in Fig. 1
and 3 for the Belousov–Zhabotinsky reaction. Even the
strong ‘‘lateral compression’’ seen in both diagrams looks
identical.
Fig. 6 (a) Isospike diagram for the autocatalator. (b) The corresponding Lyapunov phase diagram. These figures complement Fig. 7 of
Petrov et al. 7 White spines seen in the Lyapunov diagram are continuous-time generalized loci analogous to discrete-time superstable loci. 37 The
white–blue and white–black boundaries in the upper left corners are Hopf bifurcation lines. Here k = 2.5 and d = 1. Each panel displays results
obtained from the individual phase-space analysis of 1200 2 parameter points.
12196 Phys. Chem. Chem. Phys., 2011, 13, 12191–12198 This journal is c the Owner Societies 2011

Fig. 7 Top: isospike diagram displaying the Stern–Brocot organization as observed in the autocatalator after rotation in parameter space. Colors
and numbers display the number of peaks in a period of B. Some regions are too small to be identified in this scale but become visible under
magnification. Resolution: 2400 1200 parameter points. Bottom: enlargement of the white boxes, showing details of the Stern–Brocot tree. Each
of the three panels displays results obtained from the individual phase-space analysis of 1200 2 parameter points.
To minimize the lateral compression and to better visualize
the unfolding of the isospike cascade, we recomputed Fig. 6a
after suitably rotating the (m,s) parameter plane by an angle
y = p/5 around the point (m 0 ,s 0 ) = (0.0072,0.00525) according
to the transformation 37
m 0 = m 0 +(m m 0 )cos y (s s 0 )sin y, (10)
s 0 = s 0 +(m m 0 )sin y +(s s 0 )cos y. (11)
Isospike diagrams in the rotated framework are shown in
Fig. 7 where, for simplicity, we drop primes from the
transformed coordinates. This figure allows a more detailed
view of the unfolding of the cascade, which is seen to also
follow the Stern–Brocot order.
It is clear from the several phase diagrams presented here
that the hierarchical structure found for the MMOs in the
Belousov–Zhabotinsky reaction and in the autocatalator
displays an excellent agreement. In fact, macroscopically both
hierarchical structures observed may be considered as
isomorphic to each other. A detailed report about MMOs
observed in the autocatalator is being prepared and will be
presented elsewhere. 42 5. Conclusions
We performed a detailed study of the unfolding of cascades of
mixed-mode oscillations for two prototypical chemical models
by continuously recording changes in the number of spikes of
their periodic oscillations when pairs of parameters are tuned
simultaneously. In several sections of the parameter space, we
found that the number of spikes emerges organized according
to a regular tree of integers that may be easily derived from the
Stern–Brocot tree, not from the much more familiar Farey
tree. The Stern–Brocot trees are more general than Farey trees
and include them as subtrees. 29,30
The arithmetic process for generating Stern–Brocot and
Farey trees is exactly the same. But the specific tree obtained
shows sensitive dependence to the initial conditions. Alone,
the arithmetic process does not discriminate Farey from
Stern–Brocot order. In fact, since the ‘‘mediant’’ arithmetics
of both trees is exactly the same, it is generally very hard to
guess the right tree underlying cascades of oscillations based
solely on isolated cuts of parameter space or on a few
bifurcation diagrams displaying a limited number of windows.
We argued that the plain use of winding numbers based on
large/small labeling might not be a reliable indicator to
This journal is c the Owner Societies 2011 Phys. Chem. Chem. Phys., 2011, 13, 12191–12198 12197

discriminate Farey from Stern–Brocot order because such
labeling cannot sensibly be defined generically. Since only
the Farey tree is presently believed to have been sighted in
devil’s staircases generated by MMOs, an enticing challenge
now seems to be to sort out Stern–Brocot from Farey order in
such cascades. We remark that sequences of rational numbers
are easy to attribute to phenomena involving two distinct
frequencies. But it looks rather arbitrary to use the non-unique
concept of large/small peaks to emulate pairs of frequencies in
phenomena where they are not naturally present or not quite
justifiable.
The fact that the Stern–Brocot tree is clearly visible in so
many distinct sections of the parameter space of the Belousov–
Zhabotinsky model seems to imply the existence of an
exceptionally symmetric manifold in its parameter space, a
sort of ‘‘hyper-symmetrical onion’’ capable of displaying
identical-looking sections when cut along several distinct
directions, as exemplified by Fig. 1 and 3. The realistic
Belousov–Zhabotinsky model considered here is governed by
a set of fourteen parameters: 4,5 a nice and presumably timeconsuming
computational challenge now would be to try to
locate the ‘‘symmetry center’’ of such onion, if any.
Although not yet reported, we have also identified the
Stern–Brocot spike ordering to be equally present in a few
additional standard models displaying MMOs. 43 Therefore, we
believe the Stern–Brocot tree to hold great significance for the
generic description of the hierarchical structure of complex oscillatory
patterns routinely observed in chemical systems supporting
mixed-mode oscillations, as evidenced by the phase diagrams
shown above. In fact, our numerical investigations seem to
suggest the Stern–Brocot tree to emerge by far more frequently
than the nowadays so popular Farey tree: thus far we have been
unable to find evidence of Farey trees in isospike diagrams
computed for a dozen or so of the standard models known for
displaying MMOs and purportedly containing Farey trees.
In conclusion, to the abstract application of Stern in number
theory and the nice practical application devised originally by
Brocot, this paper now adds another practical use for the
Stern–Brocot tree: to describe and predict the regular and
symmetrical unfolding and multiplication of spike cascades in
mixed-mode oscillations, as indicated schematically in Fig. 4
and 5. We hope our work to motivate the experimental
verification of Stern–Brocot order of MMOs in chemical
oscillators and in other interesting systems.
JACG is grateful to R. Montandon, Buchs, Switzerland,
and P. Ribenboim, Kingston, Canada, for providing a copy of
Brocot’s delightful original paper 27 and for their kind interest
in this work. JGF thanks for a Post-Doctoral fellowship
from FCT, Portugal, grant SFRH/BPD/43608/2008, and
IF-UFRGS for hospitality. JACG is supported by CNPq,
Brazil, and by the US Air Force Office of Scientific Research,
grant FA9550-07-1-0102. All computations were done in the
computer clusters of CESUP-UFRGS.
References
1 J. Hudson, M. Hart and D. Marinko, J. Chem. Phys., 1979, 71,
1601.
2 J. S. Turner, J. C. Roux, W. D. MacCormick and H. L. Swinney,
Phys. Lett. A, 1981, 85, 9.
3 P. Ibison and S. K. Scott, J. Chem. Soc., Faraday Trans., 1990, 86,3695.
4 L. Gyo¨ rgyi and R. J. Field, Nature, 1992, 355, 808.
5 J. G. Freire, R. J. Field and J. A. C. Gallas, J. Chem. Phys., 2009,
131, 044105.
6 B. Peng, S. K. Scott and K. Showalter, J. Phys. Chem., 1990, 94, 5243.
7 V. Petrov, S. K. Scott and K. Showalter, J. Chem. Phys., 1992, 97,
6191.
8 K. Krischer, M. Eiswirth and G. Ertl, J. Chem. Phys., 1992, 96, 9161.
9 K. Krischer, M. Lu¨ bke, M. Eiswirth, W. Wolf, J. L. Hudson and
G. Ertl, Phys. D, 1993, 62, 123.
10 I. R. Epstein and K. Showalter, J. Phys. Chem., 1996, 100, 13132.
11 N. Baba and K. Krischer, Chaos, 2008, 18, 015103.
12 K. Kovacs, M. Leda, V. K. Vanag and I. R. Epstein, J. Phys.
Chem. A, 2009, 113, 146.
13 L. Yuan, Q. Gao, Y. Zhao, X. Tang and I. R. Epstein, J. Phys.
Chem. A, 2010, 114, 7014.
14 For a survey see: H. L. Swinney, Phys. D, 1983, 7, 3, and references
therein.
15 T. Braun, J. Lisboa and J. A. C. Gallas, Phys. Rev. Lett., 1992, 68,
2770, and references therein.
16 I. R. Epstein and J. A. Pojman, An Introduction to Nonlinear
Chemical Dynamics: Oscillations, Waves, Patterns, and Chaos,
Oxford University Press, New York, 1998.
17 R. J. Field and L. Gyo¨ rgyi, Chaos in Chemistry and Biochemistry,
World Scientific, Singapore, 1993.
18 P. Gray and S. K. Scott, Chemical Oscillations and Instabilities,
Oxford Science Publications, New York, 1990.
19 A. M. Zhabotinsky, Scholarpedia, 2007, 2(9), 1435.
20 L. Gyo¨ rgyi and R. J. Field, J. Phys. Chem., 1991, 95, 6594.
21 R.LarterandC.G.Steinmetz,Philos. Trans. R. Soc. London, Ser. A,
1991, 337, 291.
22 A. Milik, P. Szmolyan, H. Lo¨ ffelmann and E. Gro¨ ller, International
Journal of Bifurcation and Chaos [In Applied Sciences
and Engineering], 1998, 8, 505.
23 N. Popovic, J. Phys.: Conf. Ser., 2008, 138, 012020.
24 M. Mikikian, M. Cavarroc, L. Couëdel, Y. Tessier and
L. Boufendi, Phys. Rev. Lett., 2008, 100, 225005.
25 M. Brøns, T. J. Kaper and H. G. Rotstein, Chaos, 2008, 18, 01501.
26 M. A. Stern, J. Reine Angew. Math., 1858, 55, 193.
27 A. Brocot, Revue Chronome´trique, Journal des horlogers, Scientifique
et Pratique, 1861,3, 186.
28 S. B. Guthery, A Motif of Mathematics: History and Applications of
the Mediant and the Farey Sequence, Docent Press, Boston, 2010.
29 R. Graham, D. Knuth and O. Patashnik, Concrete Mathematics,
Addison–Wesley, 1994, 2nd edn.
30 R. Backhouse and J. F. Ferreira, Science of Computer Programming,
2011, 76, 160.
31 C. Bonatto and J. A. C. Gallas, Phys. Rev. Lett., 2008, 101, 054101.
32 G. M. Ramirez-Avila and J. A. C. Gallas, Rev. Boliv. Fis., 2008, 14,
1; G. M. Ramirez-Avila and J. A. C. Gallas, Phys. Lett. A, 2010,
375, 143.
33 J. G. Freire and J. A. C. Gallas, Phys. Rev. E: Stat., Nonlinear, Soft
Matter Phys., 2010, 82, 037202.
34 For a survey, see: J. A. C. Gallas, International Journal of
Bifurcation and Chaos, 2010, 20, 197.
35 J. A. C. Gallas, Observation of discontinuous spirals induced by
stable oscillations in a noiseless Duffing proxy, 2010, submitted for
publication.
36 J. A. C. Gallas, Phys. Rev. Lett., 1993, 70, 2714; J. A. C. Gallas,
Appl. Phys. B: Lasers Opt., 1995, 60, S203.
37 J. A. C. Gallas, Phys. A, 1994, 202, 196.
38 J. A. C. Gallas and H. E. Nusse, Journal of Economic Behavior &
Organization, 1996, 29, 447.
39 C. Bonatto, J. C. Garreau and J. A. C. Gallas, Phys. Rev. Lett.,
2005, 95, 143905.
40 C. Bonatto and J. A. C. Gallas, Phys. Rev. E: Stat., Nonlinear, Soft
Matter Phys., 2007, 75, 055204(R); C. Bonatto and J. A. C. Gallas,
Phil. Trans. R. Soc. London, 2008, 366A, 505.
41 M. A. Nascimento, J. A. C. Gallas and H. Varela, Phys. Chem.
Chem. Phys., 2011, 13, 441.
42 J. G. Freire, K. Showalter and J. A. C. Gallas, in preparation.
43 J. G. Freire and J. A. C. Gallas, Stern-Brocot trees in cascades
of mixed-mode oscillations and canards in the extended
Bonhoeffer-van der Pol and the FitzHugh-Nagumo models of
excitable systems, Phys. Lett. A, 2011, 375, 1097.
12198 Phys. Chem. Chem. Phys., 2011, 13, 12191–12198 This journal is c the Owner Societies 2011